The sum of 5 consecutive integers is 10n + 5. What is the median of the 5 integers in terms of n?

How can you prove that it must be 2n + 1?

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- Jul 25th 2012, 10:11 AMdannycFinding the median
The sum of 5 consecutive integers is 10n + 5. What is the median of the 5 integers in terms of n?

How can you prove that it must be 2n + 1? - Jul 25th 2012, 11:41 AMHallsofIvyRe: Finding the median
Call the first of the consecutive integers "i". Then the next four integers are i+1, i+2, i+3, and i+4. The sum of those five numbers is i+ (i+1)+ (i+ 3)+ (i+ 4)+ (i+ 5)= 5i+ 10= 10n+ 5. Solve that for i in terms of n. The "median" is the middle number, i+ 2.

- Jul 25th 2012, 06:14 PMdannycRe: Finding the median
How does 5i+10 imply 10n+5 ? I don't see it nor do I see how i + 2 implies 2n + 1.

- Jul 25th 2012, 09:58 PMearbothRe: Finding the median
- Jul 26th 2012, 04:07 AMdannycRe: Finding the median
Right, but I still do not know how we can set i + 2 = 2n + 1. Why can that conclusion be made algebraically?

- Jul 26th 2012, 04:55 AMearbothRe: Finding the median
1.

$\displaystyle \underbrace{i+(i+1)+\overbrace{(i+2)}^{median}+(i+ 3)+(i+4)}_{\text{according to HallsofIvy}} = \overbrace{10n+5}^{\text{according to dannyc}}$

$\displaystyle 5i+10=10n+5$

$\displaystyle 5i=10n-5=5(2n-1)$

$\displaystyle \boxed{i=2n-1}$ Now add 2 on both sides of this equation:

$\displaystyle i+\color{red}2 \color{black}= (2n-1)+\color{red}2$

$\displaystyle \boxed{i+2 = 2n+1}$ - Jul 26th 2012, 05:17 AMHallsofIvyRe: Finding the median